An additive two-level parallel variant of the DMRG algorithm with coarse-space correction

TL;DR

The paper introduces an additive two-level DMRG (A2DMRG) that replaces the traditional sequential core-by-core minimization with parallel local optimizations across TT cores and a subsequent coarse-space correction to reconcile them. This framework, inspired by additive Schwarz methods, enables parallel computation of local updates and a small, global coarse correction step, followed by TT compression to maintain low ranks. Numerical experiments on strongly correlated molecular systems demonstrate competitive convergence alongside significant parallel speedups, illustrating the method’s potential for scalable quantum-chemistry calculations. The work also outlines connections to domain decomposition and discusses future directions, including convergence analysis and adaptive coarse-space strategies.

Abstract

The density matrix renormalization group (DMRG) algorithm is a popular alternating minimization scheme for solving high-dimensional optimization problems in the tensor train format. Classical DMRG, however, is based on sequential minimization, which raises challenges in its implementation on parallel computing architectures. To overcome this, we propose a novel additive two-level DMRG algorithm that combines independent, local minimization steps with a global update step using a subsequent coarse-space minimization. Our proposed algorithm, which is directly inspired by additive Schwarz methods from the domain decomposition literature, is particularly amenable to implementation on parallel, distributed architectures since both the local minimization steps and the construction of the coarse-space can be performed in parallel. Numerical experiments on strongly correlated molecular systems demonstrate that the method achieves competitive convergence rates while achieving significant parallel speedups.

Figures (6)

• Figure 1: Convergence plot of the classical two-site DMRG and two-site A2DMRG algorithms for a linear, stretched H$_{6}$ and H$_8$ chain. The tolerances for the eigensolvers and the truncated SVD were set to $10^{-6}$ and the algorithms were run until the relative energy difference between successive half-sweeps/global iterations was less than $10^{-6}$.
• Figure 2: Convergence plot of the classical two-site DMRG and two-site A2DMRG algorithms for a linear, stretched H$_{10}$ and H$_{12}$ chain. The tolerances for the eigenvalue solvers and the truncated SVD were set to $10^{-6}$ and the algorithms were run until the relative energy difference between successive half-sweeps/global iterations was smaller than $10^{-6}$.
• Figure 3: Convergence plot of the classical DMRG and A2DMRG algorithms for the C$_2$ and N$_2$ dimers. The tolerances for the eigenvalue solvers and the truncated SVD were set to $10^{-6}$ and the algorithms were run until the relative energy difference between successive half-sweeps/global iterations was smaller than $10^{-6}$.
• Figure 4: The number of Lanczos iterations required for the micro-iterations in the classical DMRG and A2DMRG algorithms. The tolerances for the eigenvalue solvers and the truncated SVD were set to $10^{-6}$.
• Figure 5: The number of Lanczos iterations required for the second-level minimization problem in the A2DMRG algorithms. All tolerances were set to $10^{-6}$.

Theorems & Definitions (16)

• Definition 2: Separation rank of a tensor
• Definition 3: Manifold of Tensors with Fixed Separation Rank
• Definition 4: Tensor Train Parameter Space
• Definition 5: Tensor Train Contraction Mapping
• Definition 6: Orthogonal Tensor Train Decompositions
• Definition 7: Retraction Operators
• Lemma 8: Orthogonality of the One-site and Two-site Retraction Operators
• proof
• Definition 9: One-site and Two-site DMRG micro-iterations
• Lemma 10: DMRG Micro-iterations for Eigenvalue Problems